• As soon as you have comparative data, you must consider the relationships among your taxa
  • No longer an option (30+ years of methods)
• Not concerned in this class with making the tree(s)

You have a tree and data - What do I do now?

• Published trees
  • As is
  • Pruned down from published trees
• Constructed from multiple trees
• Including tree uncertainty
  • Compare effect of different taxon placements

Hard:

• Rapid speciation

Soft:

• Uncertainty of relationships
• Adjust the degrees of freedom for tests (Díaz-Uriarte and Garland 1996, 1998)
• Convert polytomies into very short branches for procedures that can't handle soft polytomies

Data from Garland et al. (1992)

• Body mass (km)
• Home range area (km²)
• "49lbr" data

1. Meet the assumptions of standard statistical tests
   • Any test you can do using "standard" methods has a phylogenetically informed counterpart.
   • What is your evolutionary model?
2. Ask and answer nuanced evolutionary questions
   • Trait evolution (continuous or categorical)
   • Trait co-evolution
   • Rates of evolution (identify nodes when rates shift significantly)
   • What is your evolutionary model?

"Real" branch lengths:

• Divergence dates (e.g., fossil calibrated)
• Genetic or character distance (relative)

Arbitrary branch lengths:

• All = 1
• Pagel's ultrametric transformation (1992): total height = deepest nesting

Either can be transformed.

• Brownian motion
• Ornstein-Uhlenbeck (OU)
• Variable rates (ACDC)
• Pagel's lambda

These models are all accomplished via branch length transformations.

• The default, null underlying model for evolution
• Simplest, most parsimonious model of drift
• No selection
• At each step, a trait value is randomly altered based on a normal distribution (mean usually 0) with some variance.

Simulate 100 Brownian motion random walks with a length of 100 steps.

set.seed(3)
nsim <- 100
t <- 0:100  # time
s2 <- 0.01

X <- foreach(i = 1:nsim, .combine = rbind) %do%
  c(0, cumsum(rnorm(n = length(t) - 1, sd = sqrt(s2))))

• Based on Ornstein & Uhlenbeck (1930)
• Brownian motion doesn't allow for adaptive evolution (drift only)
• Simplest model for an evolutionary process with selection (deterministic + stochastic)

The OU model describes the motion of a species in the phenotypic space whereby the species moves randomly within the space, but is influenced by a central tendency such that large deviations from the central optimum receive a stronger force back toward the optimum (Blomberg et al. 2003)

\[dX(t) = \alpha[\theta - X(t)]dt + \sigma dB(t)\]

• Change in trait X at time t is a function of selection \(\alpha[\theta - X(t)]dt\) and drift \(\sigma dB(t)\)
  • \(\alpha\) = strength of selection
  • \(\theta\) = optimum
  • \(\sigma\) = magnitude of drift
• \(\alpha\) and one or more local optima (\(\theta_1\), \(\theta_2\), etc.) can be estimates
• Models compared with likelihood ratio tests, AIC, etc.

Key readings:

• Hansen and Martins (1996)
• Blomberg et al. (2003)
• Butler and King (2004)

library(phytools)
set.seed(331)
tree <- pbtree(n = 15, scale = 10)

\[dX(t) = \alpha[\theta - X(t)]dt + \sigma dB(t)\]

x <- fastBM(tree,
            a = 0,           # Ancestral state at the root node
            theta = 3,       # Optimum
            alpha = 0.2,     # Strength of selection
            sig2 = 0.1,      # Variance of the BM process
            internal = TRUE) # Return states for internal nodes

phenogram(tree, x, spread.labels = TRUE, spread.cost = c(1, 0.01),
          main = "scale = 10, alpha = 0.2, sig2 = 0.1")

Rather than simulating data, estimate the values for \(\alpha\) and \(\theta\).

1. Given a tree and data (body mass in Caribbean anoles)
2. What model of evolution best fits the data?
   • Brownian motion
   • OU model with a single, clade-wide optimum
   • OU model with three body size optima (S/M/L)
   • OU model with four optima (including an ancestral size)
   • OU model based on multiple colonization events

Evolution toward two local optima:

\(\theta_{small} = 0.25\) mm head length.

• Evolution that either increases or decreases in rate over time.
  • Adaptive radiations followed by phenotypic stasis
  • Proposed by Blomberg et al. (2003)
• "Early burst" model of Harmon et al. (2010)
• Didn't really ever catch on as much as OU.

Pagel's \(\lambda\): how well does the tree fit the predicted covariance among the trait values

• \(1\) = Brownian motion
• \(>1\) = nodes pulled toward the tips (stronger covariance than expected by BM)
• \(<1\) = nodes pulled toward the root (weaker covariance than expected by BM; more common)

\[\lim_{d\to0} = \mbox{"Star phylogeny"}\]

Mesquite (http://mesquiteproject.org/)

• Enter, store, and manipulate trees
• Branch length alternations
• Models of character evolution (discrete, continuous)
• PDAP module (http://mesquiteproject.org/pdap_mesquite/)
  • Lots of functions for working with comparative data

CRAN Task View: https://cran.r-project.org/web/views/Phylogenetics.html

Common packages:

• ape
• geiger
• phytools
• picante
• caper

Lecture 11-3